Write out the first 2 columns of the matrix to the right of the 3rd column, so that you have 5 columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed). This yields:

Alternative vertical arrangement

A similar scheme based on diagonals works for 2x2 matrices:

Both are special cases of the Leibniz formula, which however does not yield similar memorization schemes for larger matrices. Sarrus' rule can also be derived by looking at the Laplace expansion of a 3×3 matrix.